Supremum and Infimum of Sets
Bounded sets always have an supremum and infimum which are generalizations of maximum and minimum.
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Study Supremum and Infimum of Sets
Definition 1 (Supremum and infimum). Let \(M\subset \mathbb{R}\) be a set.
A real number \(s\) is called the supremum of \(M\) if:
\(x\leq s\) for all \(x \in M\),
for all \(\varepsilon > 0\) there is an \(x \in M\) with \(s-\varepsilon < x\).
In this case we write \(s=\sup M\).
A real number \(l\) is called the infimum of \(M\) if:
\(x\geq l\) for all \(x \in M\),
for all \(\varepsilon > 0\) there is an \(x \in M\) with \(l + \varepsilon > x\).
In this case we write \(l=\inf M\).
We further define
\(\sup M=\infty\) if \(M\) is not bounded from above;
\(\inf M=-\infty\) if \(M\) is not bounded from below;
\(\sup \emptyset =-\infty\);
\(\inf \emptyset =\infty\).
The infimum is the greatest lower bound and the supremum is the lowest upper bound.
Example 2.
\(\sup[0,1]=1\), \(\;\;\;\inf[0,1]=0\);
\(\sup (0,1) = 1\), \(\;\;\;\;\inf (0,1) = 0\);
\(\sup\{\frac1n\;:\;n\in\mathbb{N}\}=1\), \(\qquad\qquad\;\;\;\,\inf\{\frac1n\;:\;n\in\mathbb{N}\}=0\);
\(\sup\{x\in\mathbb{Q}\;:\;x^2<2\}=\sqrt{2}\), \(\qquad\inf\{x\in\mathbb{Q}\;:\;x^2<2\}=-\sqrt{2}\);
Remark 3. Difference between sup and max (resp. inf and min) In contrast to the maximum, the supremum does not need to belong to the respective set. For instance, we have \(1=\sup(0,1)\), but \(\max(0,1)\) does not exist. The analogous statement holds true for \(\inf\) and \(\min\). However, we can make the following statement: If \(\max M\) (\(\min M\)) exists, then \(\max M=\sup M\) (\(\min M=\inf M\)).
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